Journal of Mathematics

Volume 2016 (2016), Article ID 7510567, 7 pages

http://dx.doi.org/10.1155/2016/7510567

## Herd Behavior and Financial Crashes: An Interacting Particle System Approach

Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy

Received 19 October 2015; Accepted 11 January 2016

Academic Editor: Yonghui Sun

Copyright © 2016 Vincenzo Crescimanna and Luca Di Persio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We provide an approach based on a modification of the Ising model to describe the dynamics of stock markets. Our model incorporates three different factors: imitation, the impact of external news, and private information; moreover, it is characterized by coupling coefficients, static in time, but not identical for each agent. By analogy with physical models, we consider the* temperature* parameter of the system, assuming that it evolves with memory of the past, hence considering how former news influences realized market returns. We show that a standard Ising potential assumption is not sufficient to reproduce the stylized facts characterizing financial markets; this is because it assigns low probabilities to rare events. Hence, we study a variation of the previous setting providing, also by concrete computations, new insights and improvements.

#### 1. Introduction

Empirically studying the fluctuations in the dynamics of prices for various financial assets, it has been shown that related stock returns are distributed similarly to a Gaussian random variable, but with fat tails. The latter is due to the possibilities of having large fluctuations of the prices as it has happened, for example, during recent world financial crisis; see, for example, [1–3]. One possible theoretical explanation for such phenomena relies on the so-called* econophysics* approach which states that significant fluctuations in prices correspond to collective phenomena, such as crowd effects or herd behavior; see, for example, [4], [5], [6], respectively. The main purpose of the present work is to extend those approaches which have been based on the Ising model, as, for example, in [7–9], by defining a model characterized by -potentials, such that its behavior rule, or rule of advancement, can be described by a dynamic of the Glauber-type. We notice that the latter approach allows us to build models which are easier to extend to more complex scenarios, for example, hierarchical lattice, that are the subjects of our different work, currently in progress.

In particular, our approach allows us to show that the existence of herd behavior among participants causes the fat tails property characterizing the distributions of* real* log prices. The goodness of the proposed model is analyzed by means of simulations where the stock market evolution is determined by a set of* interacting particles*, each of which represents a financial agent playing in the considered economic framework. We assume that the interactions between the aforementioned agents are driven by a potential , which is a measure of the role played by the agents in the set . Following the idea developed in [10, 11], we will consider the* temperature* of the system as a parameter describing quantities related to the probability of possible crashes. In particular we are interested in those dynamics which admit phase transitions. In fact, we can relate latter phenomenon, which is a well-defined statistical-mechanics concept, with the excess of demand represented by heavy tails. Therefore, as it happens when a physical system reaches a critical point, in financial terms we have that the system crashes if it experiences an excess of financial movements along the* same direction*; namely, a large number of single agents buy or liquidate their financial position* simultaneously*. The latter phenomenon happens in correspondence with a critical temperature value , giving rise to a phase transition.

The paper is organized as follows: in Section 2, we briefly outline the main characteristics of the so-called* econophysics approach* to the financial markets recalling two of the most relevant models used to describe the herd behavior in stock markets, namely, the Cont-Bouchaud model (see [5]) and the Sornette-Zhou model (see [8]); in Section 3, we describe our proposal, starting from the standard Ising potentials; in Section 4 we show our main theoretical results, and we also provide related numerical simulations for particularly relevant choices of the involved parameters; eventually, in Section 5, we give a* résumé* of the obtained results and we outline further lines of possible improvements and researches.

#### 2. The Econophysics Approach

During the last decades several models belonging to the* statistical-mechanics world* have been used to describe the herd effect, namely, the phenomenon which arises when a group of individuals act collectively without a centralized direction. In what follows we briefly recall the main properties of two of the most relevant approaches exploited to describe the latter type of behavior within typical* financial arenas*, namely, the Cont-Bouchaud and Sornette-Zhou models; see [5], [8], respectively.

The model developed by Cont and Bouchaud is based on the idea that the herd effect, which has been identified as one of the peculiar characteristics of stock markets as well as one of the main causes of crashes, can be described thinking that the* financial system* is constituted by random clusters of agents who imitate each other. Moreover, different clusters are allowed to take decisions independently of each other. The latter random communication structure can be described exploiting the Erdös-Rényi graph theory: see, for example, [12, 13], where with each node/agent is associated a random variable which can take the three following values: , namely, the possible agent actions. In particular, the values , , and , respectively, correspond to selling a stock, buying a stock, and to not trade at all. Concerning the edges of the graph, we have that each agent has the same probability , to be linked to another one. Such a framework can be also represented by using a potential function which admits a Gibbs measure, as shown in, for example, [14]. The latter Gibbs measure, for , being the number of agents, realizes the desired shape for the distributions of log prices; namely, they follow a Gaussian-type distribution with heavy tails. Moreover, we underline that the latter description, by analogy with the CB model, guarantees that the probability of percolation for the system is different from zero, where the percolation phenomenon, as shown in [15], is a characteristic of the interacting models that describe the stock market.

In the model proposed by Sornette and Zhou (see [8, 9]), a variant of the Ising model is exploited, allowing each agent to interact only with his* first neighbors *, and we write . Such an interaction is described by buying, , or by selling, , a single asset. It follows that, in such a model, the decision of the agent , at time , is determined by the following rule:where three different information channels are taken into account, namely, mutual influence, , external news, , and idiosyncratic judgment, . In particular is taken as a function of time: namely,Therefore, it is impossible to define a potential related to (1) in statistical-mechanics terms. More precisely, according to [16], we note that the main characteristic of the Sornette and Zhou proposal, which is the main difference with the Ising model, concerns the fluctuation of the ratio which represents, following the authors, a measure of the tendency toward imitation relative to the tendency towards idiosyncratic behavior. In the following section, we provide a way to overcome this drawback by introducing a model that can be defined using a potential.

#### 3. The Model

Let us start by considering a model with agents interacting in a network , where is the set of nodes of the square lattice (), and suppose that each node interacts with the nearest neighbor and also with agents at a maximum distance of . We underline that such a description is chosen having in mind concrete implementations. Following the idea explained in [8], we suppose that, at each time step , each agent places a buy, a sell, and order, respectively, and hence, and , respectively. The decision taken by a single agent is determined according to an interaction potential that, roughly speaking, measures the value of the agent in each set . Let us first consider the standard Ising approach, taking into account a potentialwhere the interaction parameters in the Ising case are assumed to be constants, while, in what follows, we will consider them as functions of the node .

Considering (3) with respect to a Glauber dynamic, we have that the decision of the agent at time is determined by the following* rule*:In particular, (4) highlights the following three contributions:(i)Imitation term: this is represented by the term , where is the relative propensity of the trader to be influenced by the sentiment of his nearest neighbors. We consider as a function constant in time, but not equal for all the agents. The latter assumption is due to the fact that we would give more importance to the dimension of the cluster around the agent also taking into account how single agents are affected by such interaction, exploiting the idea that not all agents are affected in the same way. In our implementation is uniformly distributed in , namely, for every , while the constants and have been empirically determined as in [8]. To be precise the constants and , respectively, define the minimum and the maximum, respectively, for the* interval of validity* of the probability distribution for the random quantity , according to the empirical tests performed with respect to the* Sornette-Zhou constant *.(ii)Impact of external news: this is represented by the term , which represents the* impact of external news* and it is assumed to be constant in time, but we allow it to be not equal for each agent. In particular, is assumed to take value as a uniform random variable in the interval , the constants being determined empirically as in [8], even if in our case describes the* maximum* concerning the news impact, while the susceptibility parameter, namely, the one indicated by in [8], has been replaced by the temperature.(iii)Idiosyncratic judgment: this is represented by the term , which embodies the idiosyncratic content of the decision of agent . It is being distributed following a Laplace distribution with mean 0 and variance 2, as a result of the Gibbs sampling used to define our dynamics.We suppose that the market price , is updated according towhere we consider, as proposed by Wyart and Bouchaud in [17], that the return rate is a linear function of the aggregate choosing of each agent at time ; in particularwhere the term measures the liquidity of the market. In our analysis, according to real market data, we have to take . In the following we will consider the distribution of returns at different time scale , and we writewhere is taken as a multiple of time step.

As highlighted by the definition of the potentials (see (3) and (11)), our dynamics is characterized by the fact that the only evolving parameter is , that is, the inverse of the temperature . The latter idea, as previously mentioned, is mainly based on the approach proposed in [10, 11], where the authors used it to describe a stock market model. In particular, in [10], it is shown that, for short time lag, that is, minute, the distribution of the returns behaves according towhere the parameter , interpreted as the* temperature of the market*, is empirically calculated as , being the market volatility. It is worth mentioning that our model provides rather different results. In particular, we have phase transitions for low temperature, while, on the contrary, in [10] the probability of financial crashes is shown to grow when the temperature increases.

By analogy with the dynamics described in [8] for the parameters , we define the following process for :which is analogous to the dynamic described in (2). In fact, miming [8], we account for the adaptive nature of agents and their learning abilities by updating the inverse of temperature. In particular, the parameter in (9) quantifies the intrinsic imitation influence of each agent in absence of other effects, and it is distributed uniformly in the interval , while the term takes into account the impact that the* external news term*, namely, , has on the decision of agents. The latter term (see [8]) is assumed to be normally distributed, ; therefore, is determined by the following rule:It is worth mentioning that we multiply the third factor for a positive constant, that is, , assuming that agents misattribute the origin of the market moves to herding rather than to impact of the news. The latter (see also [8]) confirms the idea, common to several economists, that the market follows an irrational behavior; see, for example, [18].

Following the example of Cont and Bouchaud in [5], and mainly for implementation and time consuming reasons, we consider a dynamic where agents act one at time, taking into account financial movements of the individuals that have preceded them.

Figure 1 represents the results obtained using the same values for the parameters as in [8]. In particular, Figure 1 shows that the distribution of log return follows the normal distribution; hence, we are not able to reproduce the desired* heavy tails* characteristics. Better results, in terms of adherence to real market data, cannot be obtained even considering parameters values far from those recommended by Sornette and Zhou, for example, taking or , since, doing that, we obtain unrealistic distributions.